Conformal Symmetry and Composite Operators in the $O(N{)}^3$ Tensor Field Theory

Abstract

We continue the study of the bosonic $O(N{)}^3$ model with quartic interactions and long-range propagator. The symmetry group allows for three distinct invariant ${\varphi }^4$ composite operators, known as tetrahedron, pillow and double-trace. As shown in arXiv:1903.03578 and arXiv:1909.07767, the tetrahedron operator is exactly marginal in the large-$N$ limit and for a purely imaginary tetrahedron coupling a line of real infrared fixed points (parametrized by the absolute value of the tetrahedron coupling) is found for the other two couplings. These fixed points have real critical exponents and a real spectrum of bilinear operators, satisfying unitarity constraints. This raises the question whether at large-$N$ the model is unitary, despite the tetrahedron coupling being imaginary. In this paper, we first rederive the above results by a different regularization and renormalization scheme. We then discuss the operator mixing for composite operators and we give a perturbative proof of conformal invariance of the model at the infrared fixed points by adapting a similar proof from the long-range Ising model. At last, we identify the scaling operators at the fixed point and compute the two- and three-point functions of ${\varphi }^4$ and ${\varphi }^2$ composite operators. The correlations have the expected conformal behavior and the OPE coefficients are all real, reinforcing the claim that the large-$N$ CFT is unitary.

arXiv comment44 pages, 17 figures; v2: Significant changes, one of the relevant results in version 1 has been amended

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• Long-range multi-scalar models at three loops
• Non-Perturbative Defects in Tensor Models from Melonic Trees
• The tri-fundamental quartic model
• The F-theorem in the melonic limit
• Reliability of the local truncations for the random tensor models renormalization group flow
• Remarks on a melonic field theory with cubic interaction
• Instability of complex CFTs with operators in the principal series
• Scale invariance beyond criticality within the mean-field analysis of tensorial field theories